Powers and Roots. 300. THE SQUARE OF DECIMALS. 1. The square of .5 is NOTE. A square whose side is .5 (of a linear unit) has an area of .25 (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if nec (a) Find the sum of the twelve results. 3. A square of sheet brass whose edge is .9 of a foot is what part of a square foot? 301. THE SQUARE ROOT OF DECIMALS. 1. The square root of .25 is NOTE 1.-A square whose area is .25 (of a square unit) is .5 (of a linear unit) in length. Show this by diagram. NOTE 2.-Only those decimals are perfect squares which, when in their lowest decimal terms, have numerators that are perfect squares and denominators that are perfect squares. The decimal denominators that are perfect squares are 100, 10000, 1000000, etc. 2. What is the square root of 49? √ 144 = ? √/1.44 = ? 100 Of .36? Of .64 ? √2.25 = ? √6.25 ? = (b) Find the sum of the seven results. 3. How long is the edge of a square of zinc whose area is 4.84 square feet? * *4.84 feet is 183 feet. Powers and Roots. 302. A product obtained by using a number three times as a factor is called the third power, or the cube, of the number; thus, 125 (5 × 5 × 5) is the third power, or the cube, of 5. NOTE.-One hundred twenty-five is called the third power of 5, because it may be obtained by using 5 three times as a factor. It is called the cube of 5 because it is the number of cubic units in a cube whose edge is 5 linear units. 1. Find the cube of 12; of 13; of 14;. of 15. 303. The cube root of a number is one of its three equal factors. The radical sign with a figure 3 over it indicates that the cube root of the number following it is to be taken; thus, 512, means, the cube root of 512. RULE. To find the cube root of an integral number that is a perfect cube, resolve the number into its prime factors and take one third of them as factors of the root; that is, one third as many 2's, 3's, or 5's, etc., as there are 2's, 3's, or 5's in the factors of the number. 2. Find the cube root of 1728; of 3375; of 2744; of 10648; of 5832. (b) Find the sum of the five results. Powers and Roots. 304. MISCELLANEOUS PROBLEMS. 1. Square 42. Then resolve the square of 42 into its prime factors and compare them with the prime factors of 42. 2. Cube 42. Then resolve the cube of 42 into its prime factors and compare them with the prime factors of 42. 3. Square 45. Then resolve the square of 45 into its prime factors and compare them with the prime factors of 45. 4. Cube 45. Then resolve the cube of 45 into its prime factors and compare them with the prime factors of 45. 5. Divide the cube of 15 by the square of 15. 6. Divide the cube of by the square of 3. 7. Divide the cube of .7 by the square of .7. 8. Divide the cube of 2.5 by the square of 2.5. 9. Find the square root of 5 × 5 × 7 × 7. 10. Find the cube root of 3 × 3 × 3 × 5 × 5 × 5 x 7 x 7 X 7. 11. Find the square root of each of the following perfect Algebra. 305. TO FIND THE SQUARE ROOT OF NUMBERS REPRESENTED BY LETTERS AND FIGURES. EXPLANATION. Since the square root of a number is one of its two equal factors, the square root of a*, (a × a × a × a), is a2, (a × a). The square root of a2 is a. The square root of a is a3. Let a = 3, and verify each of the foregoing statements. 1. √b = ? 2. √ a b2 = ? √bo = ? vb2 Verify. √ a2b1 = ? √ a*b* = ? Verify. Algebra. 306. TO FIND THE CUBE ROOT OF NUMBERS REPRESENTED BY LETTERS AND FIGURES. EXPLANATION. Since the cube root of a number is one of its three equal factors, the cube root of a, (a ×a×a×a×a× a), is a2, (axa). The cube root of a3 is a. The cube root of a is a3. Let a = 2, and verify each of the foregoing statements. 4. Let a 2, b = 3, and c = 5, and find the numerical = |